Welcome to MathOverflow

Question

3

1

Is it possible to find a genus two curve $C$ (over the field of complex numbers) with an endomorphism $\phi: C \to C$, such that $\phi$ has no fixed points and $\phi$ does not take any point to its hyperelliptic involution?

flag	asked Oct 9 at 17:37 xuehang 133●3

Felipe Voloch · Accepted Answer · 2012-10-09 18:18:58Z UTC

10

No, $\phi$ must come from an automorphism of $\mathbb{P}^1$ and that has fixed points.

answered Oct 9 at 18:18

Felipe Voloch
16.6k●2●36●61

thank you for your comment. Can you give a little more detail? – xuehang Oct 10 at 2:10

2

Automorphisms of hyperelliptic curves (double covers of the projective line of genus bigger than one) must come from automorphisms of the projective line. This is a standard theorem. Now, an automorphism of the projective line is a linear fractional transformation and this always has fixed points (elementary exercise). Finally a fixed point on the projective line will lift to one of the two kinds of points in your question. – Felipe Voloch Oct 10 at 2:26

genus two curve with special automorphisms

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.